Yeah, I know that the change itself is recorded, but what our department wants is people to document their process and give more information. In other words, they could change the status just to keep the ticket from being escalated or get it out of an open state without documenting their reasons for changing the status.

OK, what you want would (probably) require a feature request (via KACE User Voice forum) or would be possible using "front end ticket rules" like GillySpy's Chrome plugin demos (i.e. the "required field" script):